# A helpful tip for integration by parts

#### amilton542

Joined Nov 13, 2010
497
There comes a time and a place when a difficult integration can become an easy one, i.e.

$$F(x) = \int u\ dv = uv - \int v\;du$$

Consider,

$$F(x) = \int\;x^9cos\;ax\;dx$$

You will need to perform integration by parts TEN times in order to reduce the variable "x" and its exponent to a constant. This is (by hand) horrendous computation.

Let's integrate,

$$G(x) = \int\; x^3\;cos\;ax\;dx$$

So,

$$G(x) = \frac{x^3\;sin\;ax}{a} + \frac{(3x^2)\;cos\;ax}{a^2} - \frac{(6x)\;sin\;ax}{a^3} - \frac{(6)cos\;ax}{a^4} + C$$

Can you see the pattern?

The first term is uv, for each term there after, u is differentiated, v is integrated and the result is multiplied by minus 1 for obvious reasons in the integration by parts formula.

Given any product of the form,

$$F(x) = \int\;(a_nx^n + a_{n-1} x^{n-1}+... + a_{0}x^{0})cos\;ax\;dx$$

Where v can be a sine or cosine, can be integrated in less than one minute, provided the exponent is within reason of course.

$$F(x) = \int\;x^9cos\;ax\;dx$$